Theorem tfinds 3156

Description: Principle of Transfinite Induction (inference schema) with implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. Theorem Schema 4 of [Suppes] p. 197.

Hypotheses

Ref	Expression
tfinds.1	|- (x = (/) -> (ph <-> ps))
tfinds.2	|- (x = y -> (ph <-> ch))
tfinds.3	|- (x = suc y -> (ph <-> th))
tfinds.4	|- (x = A -> (ph <-> ta))
tfinds.5	|- ps
tfinds.6	|- (y e. On -> (ch -> th))
tfinds.7	|- (Lim x -> (A.y e. x ch -> ph))

Assertion

Ref	Expression
tfinds	|- (A e. On -> ta)

Distinct variable groups:   x,y   x,A   ch,x   ta,x   ph,y

Proof of Theorem tfinds

Step	Hyp	Ref	Expression
1		tfinds.2	. 2 |- (x = y -> (ph <-> ch))
2		tfinds.4	. 2 |- (x = A -> (ph <-> ta))
3		eloni 2953	. . . . 5 |- (x e. On -> Ord x)
4		df-lim 2948	. . . . . . . . . . . . . . . 16 |- (Lim x <-> (Ord x /\ x =/= (/) /\ x = U.x))
5	4	biimpr 152	. . . . . . . . . . . . . . 15 |- ((Ord x /\ x =/= (/) /\ x = U.x) -> Lim x)
6	5	3com23 838	. . . . . . . . . . . . . 14 |- ((Ord x /\ x = U.x /\ x =/= (/)) -> Lim x)
7	6	3expia 834	. . . . . . . . . . . . 13 |- ((Ord x /\ x = U.x) -> (x =/= (/) -> Lim x))
8	7	necon1bd 1629	. . . . . . . . . . . 12 |- ((Ord x /\ x = U.x) -> (-. Lim x -> x = (/)))
9	8	ex 373	. . . . . . . . . . 11 |- (Ord x -> (x = U.x -> (-. Lim x -> x = (/))))
10	9	com23 32	. . . . . . . . . 10 |- (Ord x -> (-. Lim x -> (x = U.x -> x = (/))))
11		orduninsuc 3109	. . . . . . . . . . 11 |- (Ord x -> (x = U.x <-> -. E.y e. On x = suc y))
12	11	biimprd 154	. . . . . . . . . 10 |- (Ord x -> (-. E.y e. On x = suc y -> x = U.x))
13	10, 12	syl5d 55	. . . . . . . . 9 |- (Ord x -> (-. Lim x -> (-. E.y e. On x = suc y -> x = (/))))
14	13	imp 350	. . . . . . . 8 |- ((Ord x /\ -. Lim x) -> (-. E.y e. On x = suc y -> x = (/)))
15	14	con1d 93	. . . . . . 7 |- ((Ord x /\ -. Lim x) -> (-. x = (/) -> E.y e. On x = suc y))
16	15	orrd 233	. . . . . 6 |- ((Ord x /\ -. Lim x) -> (x = (/) \/ E.y e. On x = suc y))
17	16	ex 373	. . . . 5 |- (Ord x -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
18	3, 17	syl 10	. . . 4 |- (x e. On -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
19		tfinds.5	. . . . . . 7 |- ps
20		tfinds.1	. . . . . . 7 |- (x = (/) -> (ph <-> ps))
21	19, 20	mpbiri 194	. . . . . 6 |- (x = (/) -> ph)
22	21	a1d 12	. . . . 5 |- (x = (/) -> (A.y e. x ch -> ph))
23		hbra1 1684	. . . . . . 7 |- (A.y e. x ch -> A.yA.y e. x ch)
24		ax-17 969	. . . . . . 7 |- (ph -> A.yph)
25	23, 24	hbim 1005	. . . . . 6 |- ((A.y e. x ch -> ph) -> A.y(A.y e. x ch -> ph))
26		raleq1 1783	. . . . . . . . . . 11 |- (x = suc y -> (A.z e. x [z / x]ph <-> A.z e. suc y[z / x]ph))
27		sbequ 1227	. . . . . . . . . . . . 13 |- (y = z -> ([y / x]ph <-> [z / x]ph))
28		ax-17 969	. . . . . . . . . . . . . 14 |- (ch -> A.xch)
29	28, 1	sbie 1194	. . . . . . . . . . . . 13 |- ([y / x]ph <-> ch)
30	27, 29	syl5bbr 533	. . . . . . . . . . . 12 |- (y = z -> (ch <-> [z / x]ph))
31	30	cbvralv 1796	. . . . . . . . . . 11 |- (A.y e. x ch <-> A.z e. x [z / x]ph)
32		ax-17 969	. . . . . . . . . . . 12 |- (ph -> A.zph)
33		hbs1 1330	. . . . . . . . . . . 12 |- ([z / x]ph -> A.x[z / x]ph)
34		sbequ12 1179	. . . . . . . . . . . 12 |- (x = z -> (ph <-> [z / x]ph))
35	32, 33, 34	cbvral 1794	. . . . . . . . . . 11 |- (A.x e. suc yph <-> A.z e. suc y[z / x]ph)
36	26, 31, 35	3bitr4g 554	. . . . . . . . . 10 |- (x = suc y -> (A.y e. x ch <-> A.x e. suc yph))
37	36	biimpd 153	. . . . . . . . 9 |- (x = suc y -> (A.y e. x ch -> A.x e. suc yph))
38		tfinds.6	. . . . . . . . . 10 |- (y e. On -> (ch -> th))
39		visset 1809	. . . . . . . . . . . 12 |- y e. V
40	39	sucid 3046	. . . . . . . . . . 11 |- y e. suc y
41	1	rcla4v 1869	. . . . . . . . . . 11 |- (y e. suc y -> (A.x e. suc yph -> ch))
42	40, 41	ax-mp 7	. . . . . . . . . 10 |- (A.x e. suc yph -> ch)
43	38, 42	syl5 21	. . . . . . . . 9 |- (y e. On -> (A.x e. suc yph -> th))
44	37, 43	sylan9r 469	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> th))
45		tfinds.3	. . . . . . . . 9 |- (x = suc y -> (ph <-> th))
46	45	adantl 388	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (ph <-> th))
47	44, 46	sylibrd 204	. . . . . . 7 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> ph))
48	47	ex 373	. . . . . 6 |- (y e. On -> (x = suc y -> (A.y e. x ch -> ph)))
49	25, 48	r19.23ai 1739	. . . . 5 |- (E.y e. On x = suc y -> (A.y e. x ch -> ph))
50	22, 49	jaoi 341	. . . 4 |- ((x = (/) \/ E.y e. On x = suc y) -> (A.y e. x ch -> ph))
51	18, 50	syl6 22	. . 3 |- (x e. On -> (-. Lim x -> (A.y e. x ch -> ph)))
52		tfinds.7	. . 3 |- (Lim x -> (A.y e. x ch -> ph))
53	51, 52	pm2.61d2 129	. 2 |- (x e. On -> (A.y e. x ch -> ph))
54	1, 2, 53	tfis3 3125	1 |- (A e. On -> ta)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 774   = wceq 954   e. wcel 956  [wsbc 1168   =/= wne 1582  A.wral 1642  E.wrex 1643  (/)c0 2276  U.cuni 2498  Ord word 2942  Oncon0 2943  Lim wlim 2944  suc csuc 2945

This theorem is referenced by:  tfindsg 3157  tfindes 3159  tfinds3 3161  oa0r 4163  om0r 4164  om1r 4167  oe1m 4169  r1tr 4634  alephon 4845  alephcard 4847  alephordi 4854

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-nul 2705  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-tr 2676  df-eprel 2827  df-po 2835  df-so 2845  df-fr 2912  df-we 2929  df-ord 2946  df-on 2947  df-lim 2948  df-suc 2949

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